| L(s) = 1 | + 2-s + 2·4-s + 3·5-s − 5·7-s + 5·8-s + 3·10-s + 6·11-s − 2·13-s − 5·14-s + 5·16-s − 2·17-s − 2·19-s + 6·20-s + 6·22-s + 5·25-s − 2·26-s − 10·28-s + 7·29-s − 3·31-s + 10·32-s − 2·34-s − 15·35-s − 2·37-s − 2·38-s + 15·40-s + 3·41-s + 7·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 4-s + 1.34·5-s − 1.88·7-s + 1.76·8-s + 0.948·10-s + 1.80·11-s − 0.554·13-s − 1.33·14-s + 5/4·16-s − 0.485·17-s − 0.458·19-s + 1.34·20-s + 1.27·22-s + 25-s − 0.392·26-s − 1.88·28-s + 1.29·29-s − 0.538·31-s + 1.76·32-s − 0.342·34-s − 2.53·35-s − 0.328·37-s − 0.324·38-s + 2.37·40-s + 0.468·41-s + 1.06·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670761 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670761 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.699669135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.699669135\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 7 T + 6 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 13 T + 98 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.53740879796961421324587228093, −9.932464615620533737295198628603, −9.655047094514461952573233080019, −9.377756947538039223846623107417, −8.895456523111674092073921309165, −8.504518780505942534690225576577, −7.43710925856592540429954935178, −7.40281787909318073610684306078, −6.77719618186618218925323984773, −6.28927169178004838764085774679, −6.28640541746606310430207161269, −5.91922621625638921535247893916, −5.01525889891267634984076146382, −4.66343261007827102154211271454, −4.01518403068575725410340197495, −3.60146969727530470879276351785, −2.88371981609774629833788006898, −2.37982520516258210570106036149, −1.81727665223344909936360619514, −1.01105507972503611508071023563,
1.01105507972503611508071023563, 1.81727665223344909936360619514, 2.37982520516258210570106036149, 2.88371981609774629833788006898, 3.60146969727530470879276351785, 4.01518403068575725410340197495, 4.66343261007827102154211271454, 5.01525889891267634984076146382, 5.91922621625638921535247893916, 6.28640541746606310430207161269, 6.28927169178004838764085774679, 6.77719618186618218925323984773, 7.40281787909318073610684306078, 7.43710925856592540429954935178, 8.504518780505942534690225576577, 8.895456523111674092073921309165, 9.377756947538039223846623107417, 9.655047094514461952573233080019, 9.932464615620533737295198628603, 10.53740879796961421324587228093
